Theorem cvlatcvr1 30213

Description: An atom is covered by its join with a different atom. (Contributed by NM, 5-Nov-2012.)

Hypotheses

Ref	Expression
cvlatcvr1.j	|- .\/ = ( join ` K )
cvlatcvr1.c	|- C = ( <o ` K )
cvlatcvr1.a	|- A = ( Atoms ` K )

Assertion

Ref	Expression
cvlatcvr1	|- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> P C ( P .\/ Q ) ) )

Proof of Theorem cvlatcvr1

Step	Hyp	Ref	Expression
1		simp13 990	. . . 4 |- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> K e. CvLat )
2		cvlatl 30197	. . . 4 |- ( K e. CvLat -> K e. AtLat )
3	1, 2	syl 16	. . 3 |- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> K e. AtLat )
4		eqid 2438	. . . 4 |- ( meet ` K ) = ( meet ` K )
5		eqid 2438	. . . 4 |- ( 0. ` K ) = ( 0. ` K )
6		cvlatcvr1.a	. . . 4 |- A = ( Atoms ` K )
7	4, 5, 6	atnem0 30190	. . 3 |- ( ( K e. AtLat /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> ( P ( meet ` K ) Q ) = ( 0. ` K ) ) )
8	3, 7	syld3an1 1231	. 2 |- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> ( P ( meet ` K ) Q ) = ( 0. ` K ) ) )
9		eqid 2438	. . . 4 |- ( Base ` K ) = ( Base ` K )
10	9, 6	atbase 30161	. . 3 |- ( P e. A -> P e. ( Base ` K ) )
11		cvlatcvr1.j	. . . 4 |- .\/ = ( join ` K )
12		cvlatcvr1.c	. . . 4 |- C = ( <o ` K )
13	9, 11, 4, 5, 12, 6	cvlcvrp 30212	. . 3 |- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. ( Base ` K ) /\ Q e. A ) -> ( ( P ( meet ` K ) Q ) = ( 0. ` K ) <-> P C ( P .\/ Q ) ) )
14	10, 13	syl3an2 1219	. 2 |- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( ( P ( meet ` K ) Q ) = ( 0. ` K ) <-> P C ( P .\/ Q ) ) )
15	8, 14	bitrd 246	1 |- ( ( ( K e. OML /\ K e. CLat /\ K e. CvLat ) /\ P e. A /\ Q e. A ) -> ( P =/= Q <-> P C ( P .\/ Q ) ) )

Syntax hints:   -> wi 4   <-> wb 178   /\ w3a 937   = wceq 1653   e. wcel 1726   =/= wne 2601   class class class wbr 4215   ` cfv 5457  (class class class)co 6084   Basecbs 13474   joincjn 14406   meetcmee 14407   0.cp0 14471   CLatccla 14541   OMLcoml 30047   <o ccvr 30134   Atomscatm 30135   AtLatcal 30136   CvLatclc 30137

This theorem is referenced by:  cvlatcvr2  30214  atcvr1  30288

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-rep 4323  ax-sep 4333  ax-nul 4341  ax-pow 4380  ax-pr 4406  ax-un 4704

This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2712  df-rex 2713  df-reu 2714  df-rab 2716  df-v 2960  df-sbc 3164  df-csb 3254  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-pw 3803  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-iun 4097  df-br 4216  df-opab 4270  df-mpt 4271  df-id 4501  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-res 4893  df-ima 4894  df-iota 5421  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-fv 5465  df-ov 6087  df-oprab 6088  df-mpt2 6089  df-1st 6352  df-2nd 6353  df-undef 6546  df-riota 6552  df-poset 14408  df-plt 14420  df-lub 14436  df-glb 14437  df-join 14438  df-meet 14439  df-p0 14473  df-lat 14480  df-clat 14542  df-oposet 30048  df-ol 30050  df-oml 30051  df-covers 30138  df-ats 30139  df-atl 30170  df-cvlat 30194